JODS-PD: Single-Layer Primal–Dual Dynamical Optimization for IRS-Assisted NOMA Beamforming

Joint active and passive beamforming in an intelligent reflecting surface (IRS)-assisted non-orthogonal multiple-access (NOMA) downlink is a non-convex constrained problem whose Karush–Kuhn–Tucker (KKT) points are generally non-isolated. We formulate the normalized beamformers, continuous IRS angles, and inequality multipliers as one coupled projected primal–dual dynamical state. Here, single-layer means simultaneous primal–dual evolution; the numerical integrator may still contain multiple stages. Angle coordinates remove the unit-modulus equalities, while an adjacent-chain representation reduces the received-power ordering constraints without changing their feasible set for a fixed decoding order. We prove equilibrium–KKT equivalence and local transverse exponential attraction of the KKT equilibrium set under strict complementarity, locally constant active rank, nonzero streams, a quotient second-order condition, and a sufficiently large finite penalty parameter. A stiff-ODE warm-up followed by a nonmonotone semismooth pseudo-transient continuation (PTC) method computes a steady-state root.